Use the given functions to find each of the following and their respective domains. Do not try to simplify the resulting functions and be sure to write each domain using interval notation.
f(x)=4/x and g(x)=x+9
(f+g)(x)=
d=
(f-g)(x)=
d=
(fg)(x)=
d=
(f/g)(x)=
d=
To find each of the following and their respective domains, we will evaluate the given functions as indicated:
1) (f+g)(x) = f(x) + g(x)
= (4/x) + (x+9)
= 4/x + x + 9
Therefore, (f+g)(x) = 4/x + x + 9
Domain (d): The domain of (f+g)(x) is all real numbers except for x = 0 (since the function f(x) = 4/x has a denominator of x).
Domain: (-∞, 0) ∪ (0, ∞)
2) (f-g)(x) = f(x) - g(x)
= (4/x) - (x+9)
= 4/x - x - 9
Therefore, (f-g)(x) = 4/x - x - 9
Domain (d): The domain of (f-g)(x) is all real numbers except for x = 0 (since the function f(x) = 4/x has a denominator of x).
Domain: (-∞, 0) ∪ (0, ∞)
3) (fg)(x) = f(x) * g(x)
= (4/x) * (x+9)
= 4(x+9)/x
Therefore, (fg)(x) = 4(x+9)/x
Domain (d): The domain of (fg)(x) is all real numbers except for x = 0 (since the function f(x) = 4/x has a denominator of x).
Domain: (-∞, 0) ∪ (0, ∞)
4) (f/g)(x) = f(x) / g(x)
= (4/x) / (x+9)
= 4 / (x(x+9))
Therefore, (f/g)(x) = 4 / (x(x+9))
Domain (d): The domain of (f/g)(x) is all real numbers except for x = 0 and x = -9 (since the functions f(x) = 4/x and g(x) = x+9 both have a denominator of x and x+9, respectively).
Domain: (-∞, -9) ∪ (-9, 0) ∪ (0, ∞)
To find each of the given functions and their respective domains, we need to perform the specified operations on the given functions and determine the resulting functions and their domains.
1. (f+g)(x) = f(x) + g(x)
= (4/x) + (x + 9)
= 4/x + x + 9
The domain of (f+g)(x) is the intersection of the domains of f(x) and g(x). The domain of f(x) is all real numbers except x = 0 (since division by zero is undefined). The domain of g(x) is all real numbers. Therefore, the domain of (f+g)(x) is all real numbers except x = 0.
Domain: (-∞, 0) ∪ (0, +∞)
2. (f-g)(x) = f(x) - g(x)
= (4/x) - (x + 9)
= 4/x - x - 9
The domain of (f-g)(x) is also the intersection of the domains of f(x) and g(x). The domain of f(x) is all real numbers except x = 0, and the domain of g(x) is all real numbers. Therefore, the domain of (f-g)(x) is all real numbers except x = 0.
Domain: (-∞, 0) ∪ (0, +∞)
3. (fg)(x) = f(x) * g(x)
= (4/x) * (x + 9)
= 4(x + 9)/x
The domain of (fg)(x) is the intersection of the domains of f(x) and g(x), which is all real numbers except x = 0.
Domain: (-∞, 0) ∪ (0, +∞)
4. (f/g)(x) = f(x) / g(x)
= (4/x) / (x + 9)
= (4/x) * (1/(x + 9))
= 4/(x(x + 9))
The domain of (f/g)(x) is determined by the restrictions on the domains of f(x) and g(x). The domain of f(x) is all real numbers except x = 0, and the domain of g(x) is all real numbers. Therefore, the domain of (f/g)(x) is all real numbers except x = 0.
Domain: (-∞, 0) ∪ (0, +∞)
To find each of the given functions and their respective domains, we will perform the specified operations using the given functions and then determine the resulting domains.
1. (f+g)(x):
To find (f+g)(x), we need to add the functions f(x) and g(x).
(f+g)(x) = f(x) + g(x) = 4/x + (x + 9)
To determine the domain, we need to consider the restrictions on x that would result in undefined values. In this case, the function f(x) = 4/x is undefined when x = 0, as division by zero is not allowed. Thus, the domain for (f+g)(x) is all real numbers except 0.
Domain: (-∞, 0) U (0, ∞)
2. (f-g)(x):
To find (f-g)(x), we need to subtract the function g(x) from f(x).
(f-g)(x) = f(x) - g(x) = 4/x - (x + 9)
Similar to the previous case, the function f(x) = 4/x is undefined when x = 0. Therefore, the domain for (f-g)(x) is the same as the previous case.
Domain: (-∞, 0) U (0, ∞)
3. (fg)(x):
To find (fg)(x), we need to multiply the functions f(x) and g(x).
(fg)(x) = f(x) * g(x) = (4/x) * (x + 9)
Since multiplication is defined for all real numbers, there are no additional restrictions on the domain. Thus, the domain for (fg)(x) is also all real numbers.
Domain: (-∞, ∞)
4. (f/g)(x):
To find (f/g)(x), we need to divide the function f(x) by g(x).
(f/g)(x) = f(x) / g(x) = (4/x) / (x + 9)
For this case, we need to consider the restrictions on x that would result in a denominator equal to zero, as division by zero is not allowed. In the denominator, (x + 9) becomes zero when x = -9. Therefore, the domain for (f/g)(x) excludes this value.
Domain: (-∞, -9) U (-9, ∞)
To summarize:
(f+g)(x) = 4/x + (x + 9), with domain: (-∞, 0) U (0, ∞)
(f-g)(x) = 4/x - (x + 9), with domain: (-∞, 0) U (0, ∞)
(fg)(x) = (4/x) * (x + 9), with domain: (-∞, ∞)
(f/g)(x) = (4/x) / (x + 9), with domain: (-∞, -9) U (-9, ∞)